Of course, we can apply the formulas from [1]}, [2]}, [3]}, [4]} directly to (REF ), but this leads to rather cumbersome expressions that are not explicitly covariant because of the \((3+1)\)  decomposition. Therefore, we reduce our 3-dimensional problem to the 4-dimensional one preserving explicitly the general covariance. To this end, we introduce the auxiliary integration variable,
\(e^{-\tau H(\omega )}=\int _{-\infty }^{\infty }d p_0 e^{-\tau H(p_0)}\delta (p_0-\omega )=\int _{-\infty }^{\infty }d p_0 e^{-\tau [H(p_0)+\frac{1}{\xi ^2}(\omega ^2-p^2_0)]}\delta (p_0-\omega ),\) 
